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In this talk I will present some recent results on the long time behaviour of the overdamped Langevin dynamics for Brownian particles moving in a multiscale, rugged energy landscape. The dynamics of such processes can be quite complicated, in particular in the low temperature regime, since metastable states, corresponding to local minima of the potential, can (co-)exist at all scales. We will show how we can obtain a coarse-grained description for the dynamics at large scales, given by a stochastic differential equation with multiplicative noise, despite the fact that the noise in the original dynamics is additive. We then show that the combined effect of noise and multiscale structure leads to hysteresis effects in the bifurcation diagram for the equilibrium coarse-grained dynamics. In the second part of the talk I will present recent results on the mean field limit of systems of interacting diffusions in a multiscale confining potential. The mean field limit is described by a nonlinear, nonlocal Fokker-Planck equation of McKean-Vlasov type that exhibits phase transitions. The effect of the multiscale structure of the potential on the phase diagram will be discussed in detail.
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